In mathematics, the one-point compactification (or Alexandroff compactification) is a compactification of a topological space X, found by P. S. Alexandroff (1924). In mathematics, compactification is the process or result of enlarging a topological space to make it compact. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Pavel Sergeevich Alexandrov (Па́вел Серге́евич Алекса́ндров, sometimes romanized Alexandroff or Aleksandrov) (born May 7, 1896 - died November 16, 1982) was a Russian mathematician. ...

The one-point compactification αX of X has as underlying set the disjoint union of X and an extra point, usually written as ∞. The open sets not containing ∞ are the open sets of X. The open sets containing ∞ are the complements of closed compact subsets of X. In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

The one-point compactification is compact and contains X as an open subspace. It is a Hausdorff space if and only if X is locally compact Hausdorff. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

See also

• Stone–Čech compactification
• Wallman compactification

In mathematics, the Stone–Čech compactification of a Tychonoff topological space is the largest Hausdorff compactification of , in the sense that any Hausdorff compactification of is a quotient of in a way that preserves the embeddings of . ...

References

• Fedorchuk, V.V. (2001), “Aleksandrov compactification”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Alexandroff, P.S. (1924), “Über die Metrisation der im Kleinen kompakten topologischen Räume”, Math. Ann. 92: 294–301, DOI 10.1007/BF01448011